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Properties

Label 2-1960-1960.219-c0-0-1
Degree $2$
Conductor $1960$
Sign $-0.934 - 0.355i$
Analytic cond. $0.978167$
Root an. cond. $0.989023$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0747 − 0.997i)2-s + (−0.988 − 0.149i)4-s + (−0.733 − 0.680i)5-s + (−0.222 − 0.974i)7-s + (−0.222 + 0.974i)8-s + (0.826 − 0.563i)9-s + (−0.733 + 0.680i)10-s + (−1.21 − 0.825i)11-s + (1.78 − 0.858i)13-s + (−0.988 + 0.149i)14-s + (0.955 + 0.294i)16-s + (−0.5 − 0.866i)18-s + (−0.955 + 1.65i)19-s + (0.623 + 0.781i)20-s + (−0.914 + 1.14i)22-s + (−0.365 − 0.930i)23-s + ⋯
L(s)  = 1  + (0.0747 − 0.997i)2-s + (−0.988 − 0.149i)4-s + (−0.733 − 0.680i)5-s + (−0.222 − 0.974i)7-s + (−0.222 + 0.974i)8-s + (0.826 − 0.563i)9-s + (−0.733 + 0.680i)10-s + (−1.21 − 0.825i)11-s + (1.78 − 0.858i)13-s + (−0.988 + 0.149i)14-s + (0.955 + 0.294i)16-s + (−0.5 − 0.866i)18-s + (−0.955 + 1.65i)19-s + (0.623 + 0.781i)20-s + (−0.914 + 1.14i)22-s + (−0.365 − 0.930i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $-0.934 - 0.355i$
Analytic conductor: \(0.978167\)
Root analytic conductor: \(0.989023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :0),\ -0.934 - 0.355i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7421966515\)
\(L(\frac12)\) \(\approx\) \(0.7421966515\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0747 + 0.997i)T \)
5 \( 1 + (0.733 + 0.680i)T \)
7 \( 1 + (0.222 + 0.974i)T \)
good3 \( 1 + (-0.826 + 0.563i)T^{2} \)
11 \( 1 + (1.21 + 0.825i)T + (0.365 + 0.930i)T^{2} \)
13 \( 1 + (-1.78 + 0.858i)T + (0.623 - 0.781i)T^{2} \)
17 \( 1 + (0.733 + 0.680i)T^{2} \)
19 \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.365 + 0.930i)T + (-0.733 + 0.680i)T^{2} \)
29 \( 1 + (0.222 - 0.974i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.63 - 0.246i)T + (0.955 - 0.294i)T^{2} \)
41 \( 1 + (0.0332 - 0.145i)T + (-0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (-0.0546 + 0.728i)T + (-0.988 - 0.149i)T^{2} \)
53 \( 1 + (0.722 + 0.108i)T + (0.955 + 0.294i)T^{2} \)
59 \( 1 + (-0.326 + 0.302i)T + (0.0747 - 0.997i)T^{2} \)
61 \( 1 + (-0.955 + 0.294i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (0.988 - 0.149i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (-1.03 + 0.702i)T + (0.365 - 0.930i)T^{2} \)
97 \( 1 - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.810375268671154137840887833694, −8.282780961784070629661467792034, −7.78784869548067765973233771205, −6.41901768692884608512538168895, −5.56977625187679975167025913557, −4.52857463467434597721411774010, −3.67213409052099819372134629921, −3.41186134193345065387762718787, −1.59204853405602646074165138011, −0.55375238537567859369689011395, 2.05444645754324081209956134039, 3.31261501664084794861543047146, 4.29035924587682725835404829352, 4.97826577629306245220804100387, 5.99332177436876567915482297065, 6.79395947623489539476061059657, 7.31544466653034638617719848251, 8.187368335387522098140219097621, 8.777603992480431372493546584994, 9.588148468465758752830043369652

Graph of the $Z$-function along the critical line